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Precalculus : Derivatives

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Example Questions

Example Question #31 : Find The First Derivative Of A Function

Find the derivative: f(x)=6x^4+2x^3-x^2+5

Possible Answers:

f'(x)=24x^3+6x^2-2x+5x

Answer is not present.

f'(x)=6x^3+3x^2-2x+5

f'(x)=24x^3+6x^2-2x

f'(x)=6x^3+3x^2-2x

Correct answer:

f'(x)=24x^3+6x^2-2x

Explanation:

f(x)=6x^4+2x^3-x^2+5

The derivative of this function, known as f'(x) (read as f prime of x), can be found by using the Power rule of derivatives on each term in the function. Power rule: $x^n=nx^{n-1}$ . If there is a value in front of x we multiply it by the "n" we carried over. For example taking the derivative of 2x^4=8x^3 .

Using the Power rule:

f'(x)=24x^3+6x^2-2x

Note that the derivative of any constant is simply 0.

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Example Question #32 : Find The First Derivative Of A Function

Find the first derivative of the function

$f(x)=x^{2}cos(2x)$

Possible Answers:

$f'(x)=2xcos(2x)-2x^{2}sin(2x)$

$f'(x)=-2xsin(2x)-x^{2}cos(2x)$

f'(x)=2x-2sin(2x)

$f'(x)=x^{2}cos(2x)-2sin(2x)$

f'(x)=-4xsin(2x)

Correct answer:

$f'(x)=2xcos(2x)-2x^{2}sin(2x)$

Explanation:

Use the product rule to find the first derivative.

h(x)=f(x)g(x)

h'(x)=f'(x)g(x)+g'(x)f(x)

Here, let $f(x)=x^{2}$ and g(x)=cos(2x)

$h'(x)=(x^{2})'cos(2x)+x^{2}(cos(2x))'$

$h'(x)=2xcos(2x)+(-2x^{2}sin(2x))$

$h'(x)=2xcos(2x)-2x^{2}sin(2x)$

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Example Question #32 : Find The First Derivative Of A Function

Find the first derivative of the function

$f(x)=\sqrt{x^{2}+4x}$ .

Possible Answers:

$f'(x)=(2x+4)\sqrt{x^{2}+4x}$

$f'(x)=\frac{2x+4}\sqrt{x^{2}+4x}{}$

$f'(x)=\frac{x+2}{\sqrt{x^{2}+4x}}$

$f'(x)=\frac{1}{2\sqrt{x^{2}+4x}}$

$f'(x)=2x\sqrt{x+4}$

Correct answer:

$f'(x)=\frac{x+2}{\sqrt{x^{2}+4x}}$

Explanation:

Using the power rule, $f(x)=\sqrt{x}=x^{\frac{1}{2}}$

$f'(x)=\frac{1}{2}x^{-\frac{1}{2}}=\frac{1}{2\sqrt{x}}$

Apply the Chain Rule.

$f'(x)=\frac{1}{2\sqrt{x^{2}+4x}}(x^{2}+4x)'$

$f'(x)=\frac{1}{2\sqrt{x^{2}+4x}}(2x+4)$

$f'(x)=\frac{x+2}{\sqrt{x^{2}+4x}}$

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Example Question #34 : Find The First Derivative Of A Function

f(x)=(x^4-2x-1) and g(x)=3x-5 . Find the derivative of h(x)=f(x)g(x) .

Possible Answers:

f'(x)=15x^4-20x^2-12x+7

f'(x)=15x^4-20x^3+7

f'(x)=15x^3-20x^3+12x+7

f'(x)=15x^4-20x^3-12x+7

f'(x)=15x^4-20x^3-12x-7

Correct answer:

f'(x)=15x^4-20x^3-12x+7

Explanation:

Since we have the product of two different expressions we will use the product rule: f'(x)g(x)+f(x)g'(x) .

(x^4-2x-1)(3x-5)

Using the product rule:

(4x^3-2)(3x-5)+(x^4-2x-1)*3

Combine and simplify:

12x^4-20x^3-6x+10+3x^4-6x-3

h'(x)=15x^4-20x^3-12x+7

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Example Question #35 : Find The First Derivative Of A Function

What is the derivative of the following function:

y=tan(x)-2cos(x)

with respect to .

Possible Answers:

sec^2(t)+2sin(t)

$(sec^2(x)+2sin(x))\cdot t$

(tan(x)-2cos(x))t

Correct answer:

Explanation:

This question is meant to test your understanding of derivatives.

Since the term tan(x)-2cos(x) is completely unrelated to the variable ,

$\frac{\mathrm{d} }{\mathrm{d} t}(tan(x)-2cos(x))=0$ .

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Example Question #31 : Find The First Derivative Of A Function

Determine the first derivative of y=2x^5 .

Possible Answers:

7x^4

10x^5

10x

7x^5

10x^4

Correct answer:

10x^4

Explanation:

Use the power rule to solve this derivative.

$\frac{d}{dx} = x^n= nx^{n-1}$

$y'=5\cdot 2x^{5-1} = 10x^4$

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Example Question #37 : Find The First Derivative Of A Function

Calculate the derivative of $y= 5\pi + 2$ .

Possible Answers:

$\frac{2}{5}$

$\frac{5}{2}$

Correct answer:

Explanation:

The derivative of constants, or numbers, are zero. The term $5\pi + 2$ has no variables and will simplify into some number if the $\pi$ was replaced with 3.14 .

The result of $5\pi + 2$ will be a constant. Therefore, the derivative, or slope, of $y= 5\pi + 2$ is zero.

The answer is .

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Example Question #38 : Find The First Derivative Of A Function

Find the derivative of: y=(2x+2)^2

Possible Answers:

4x+4

8x+8

4x+8

8x+4

8x+12

Correct answer:

8x+8

Explanation:

To find the derivative of this function, use the power rule and the chain rule.

The power rule is: $\frac{d}{dx} x^n = nx^{n-1}$

The chain rule is to take the derivative of the inner function.

Apply the power rule for the function and the chain rule. The derivative of 2x+2 is .

$y'=2(2x+2) \cdot 2 = 4(2x+2) = 8x+8$

The answer is: 8x+8

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Example Question #32 : Find The First Derivative Of A Function

Find the first derivative of .

$f(x)=\sin(x)+3x^2-1$

Possible Answers:

$\cos(x)+6x$

$\sin(x)+6x$

$-\cos(x)+6x$

$-\cos(x)+3x^2$

Correct answer:

$\cos(x)+6x$

Explanation:

To simplify, simply take the derivative according to the rules for derivatives. Thus,

$f'(x)=(\sin(x)+3x^2-1)'=\cos(x)+6x$

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Example Question #132 : Introductory Calculus

Find the first derivative of the following function:

$f(x)=6x-\ln{x}$

Possible Answers:

$6+\frac{1}{x}$

all

$6-\frac{1}{x}$

undefined

Correct answer:

$6-\frac{1}{x}$

Explanation:

To solve, simply use the power rule and rule for differentiating natural log as outlined below.

Power rule: $(x^n)'=nx^{n-1}$

Differentiating natural log: $(\ln(x))'=\frac{1}{x}*x'$

Thus,

$f(x)=6x-\ln{x}$

$f'(x)=6x^{1-1}-\frac{1}{x}*1=6x-\frac{1}{x}$

Thus, our first derivative is:

$f'(x)=6x-\frac{1}{x}$

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Precalculus : Derivatives

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #31 : Find The First Derivative Of A Function

Example Question #32 : Find The First Derivative Of A Function

Example Question #32 : Find The First Derivative Of A Function

Example Question #34 : Find The First Derivative Of A Function

Example Question #35 : Find The First Derivative Of A Function

Example Question #31 : Find The First Derivative Of A Function

Example Question #37 : Find The First Derivative Of A Function

Example Question #38 : Find The First Derivative Of A Function

Example Question #32 : Find The First Derivative Of A Function

Example Question #132 : Introductory Calculus

All Precalculus Resources

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